After further research into the correspondence between (countable) ordinals and functions, I believe I have discovered the function that represents a growth rate corresponding to the Feferman-Schütte ordinal . The details are here:

Interestingly enough, the function turns out to be related to transforming binary trees. In particular, the diagonalized function E(n) involves transforming a left-branching linear chain to a right-branching linear chain, and in the process adding many nodes... MANY nodes (so many that I suspect it's not possible to describe using any other known notation for large numbers). All this is done by a very simple-looking two-step transform, applied repeatedly until the tree becomes a right-branching chain.

Anyway, let me know what y'all think.

P.S. Oh, and I should add that even though I derived the function based on the structure of transfinite ordinals, the function itself has no direct connection with ordinals. It's really just a binary tree transforming function. (Just so the anti-Cantorians here don't write it off as uninteresting. )